To allow the acquisition of the deductive-logic method and provide basic mathematical tools for the differential and integral calculus. Each topic will be strictly introduced and treated by carrying out, whenever needed, detailed demonstrations and by referring largely to the physical meaning, the geometrical interpretation and the numerical application. A proper methodology combined with a reasonable skill in the use of the concepts and results of the integro-differential calculus, will enable students to face more applicative concepts that will be tackled during the succeeding courses.
Canali
teacher profile teaching materials
Calcolo of P. Marcellini and C. Sbordone.
1) Real numbers and functions
Natural, whole and rational numbers; density of rationals (5). Axioms of real numbers (2). Overview of set theory (4).
The intuitive concept of function (6) and Cartesian representation (7).
Injective, surjective, bijective and invertible functions. Monotonic functions (8).
Absolute value (9). The principle of induction (13).
2) Complements to real numbers
Maximum, minimum, supremum, infimum.
7) Succession limits
Definition and first properties (56.57).
Limited successions (58). Operations with limits (59).
Indefinite forms (60).
Comparative theorems (61). Other properties of succession limits (62).
Notable limits (63). Monotone sequences, the number e (64).
Sequence going to infinity of increasing order (67).
8) Function limits. Continuous functions
Definition of limit and property (71,72,73).
Continuous functions (74). discontinuity (75).
Theorems on continuous functions (76).
9) Additions to the limits
The theorem on monotonic sequences (80).
Extracted successions; the Bolzano-Weierstrass theorem (81).
The Weierstrass theorem (82).
Continuity of monotonic functions and inverse functions (83).
10) Derivatives
Definition and physical meaning (88-89). Operations with derivatives (90).
Derivatives of compound functions and inverse functions (91).
Derivative of elementary functions (92).
Geometric meaning of the derivative: tangent line (93).
11) Applications of derivatives. Function study
Maximum and minimum relative. Fermat's theorem (95).
Theorems of Rolle and Lagrange (96).
Increasing, decreasing, convex and concave functions (97-98).
De l'Hopital theorem (99).
Study of the graph of a function (100).
Taylor's formula: first properties (101).
14) Integration according to Riemann
Definition (117). Properties of the defined integrals (118).
Uniform continuity. Cantor's theorem (119).
Integrability of continuous functions (120).
The theorems of the average (121).
15) Undefined integrals
The fundamental theorem of integral calculus (123).
Primitives (124). The indefinite integral (125). Integration by parts and by substitution
(126,127,128,129).
Improper integrals (132).
16) Taylor's formula
Rest of Peano (135).
Use of Taylor's formula in the calculation of limits (136).
17) Series
Numerical series (141).
Series with positive terms (142).
Geometric series and harmonic series (143.144).
Convergence criteria (145).
Alternate series (146).
Absolute convergence (147).
Taylor series (149).
Programme
The numbers refer to the chapters and paragraphs of the textbook:Calcolo of P. Marcellini and C. Sbordone.
1) Real numbers and functions
Natural, whole and rational numbers; density of rationals (5). Axioms of real numbers (2). Overview of set theory (4).
The intuitive concept of function (6) and Cartesian representation (7).
Injective, surjective, bijective and invertible functions. Monotonic functions (8).
Absolute value (9). The principle of induction (13).
2) Complements to real numbers
Maximum, minimum, supremum, infimum.
7) Succession limits
Definition and first properties (56.57).
Limited successions (58). Operations with limits (59).
Indefinite forms (60).
Comparative theorems (61). Other properties of succession limits (62).
Notable limits (63). Monotone sequences, the number e (64).
Sequence going to infinity of increasing order (67).
8) Function limits. Continuous functions
Definition of limit and property (71,72,73).
Continuous functions (74). discontinuity (75).
Theorems on continuous functions (76).
9) Additions to the limits
The theorem on monotonic sequences (80).
Extracted successions; the Bolzano-Weierstrass theorem (81).
The Weierstrass theorem (82).
Continuity of monotonic functions and inverse functions (83).
10) Derivatives
Definition and physical meaning (88-89). Operations with derivatives (90).
Derivatives of compound functions and inverse functions (91).
Derivative of elementary functions (92).
Geometric meaning of the derivative: tangent line (93).
11) Applications of derivatives. Function study
Maximum and minimum relative. Fermat's theorem (95).
Theorems of Rolle and Lagrange (96).
Increasing, decreasing, convex and concave functions (97-98).
De l'Hopital theorem (99).
Study of the graph of a function (100).
Taylor's formula: first properties (101).
14) Integration according to Riemann
Definition (117). Properties of the defined integrals (118).
Uniform continuity. Cantor's theorem (119).
Integrability of continuous functions (120).
The theorems of the average (121).
15) Undefined integrals
The fundamental theorem of integral calculus (123).
Primitives (124). The indefinite integral (125). Integration by parts and by substitution
(126,127,128,129).
Improper integrals (132).
16) Taylor's formula
Rest of Peano (135).
Use of Taylor's formula in the calculation of limits (136).
17) Series
Numerical series (141).
Series with positive terms (142).
Geometric series and harmonic series (143.144).
Convergence criteria (145).
Alternate series (146).
Absolute convergence (147).
Taylor series (149).
Core Documentation
S. Lang, A First Course in Calculus, Springer Ed.Type of delivery of the course
8 hours of frontal teaching a week In the event of an extension of the health emergency from COVID-19, all the provisions that regulate the methods of carrying out the teaching activities will be implemented. In particular, the following methods will apply: live remote lesson and recording of the lesson itself.Attendance
not compulsoryType of evaluation
written test and subsequent oral test teacher profile teaching materials
P. Marcellini, C. Sbordone: Esercitazioni di matematica (Vol. 1/1). Ed. Liguori, 2016
P. Marcellini, C. Sbordone: Esercitazioni di matematica (Vol. 1/2). Ed. Liguori, 2016
Programme
Exercises on base topics in Calculus (sequences, limits of real functions, derivatives, integrals, numerical series, power series, complex numbers) and applications.Core Documentation
P. Marcellini, C. Sbordone: Analisi Matematica (vol. 1). Ed. Liguori, 2015P. Marcellini, C. Sbordone: Esercitazioni di matematica (Vol. 1/1). Ed. Liguori, 2016
P. Marcellini, C. Sbordone: Esercitazioni di matematica (Vol. 1/2). Ed. Liguori, 2016
Type of delivery of the course
Written test and oral exam about theoryType of evaluation
Evaluation of the student's ability to solve the proposed problems and the exposition of the theory teacher profile teaching materials
Sequences, definition of limit, operations with limits, comparison theorems, infinitives of increasing order.
Limits of function, continuity, link with the limits of sequences, theorems on continuous functions.
Derivatives, geometric meaning, theorems on differentiable functions, relative maximums and minimums, applications to the study of functions. Indefinite integrals, integration by parts and by substitution, definite integrals, fundamental theorem of integral calculus, improper integrals. Numerical series, simple and absolute convergence, convergence criteria.
Series of functions, point and total convergence. Complex numbers.
Marcellini, Sbordone - Exercises in mathematics Vol. 1, 2.
Programme
Real numbers and functions, set theory, induction principle, infimum and supremum.Sequences, definition of limit, operations with limits, comparison theorems, infinitives of increasing order.
Limits of function, continuity, link with the limits of sequences, theorems on continuous functions.
Derivatives, geometric meaning, theorems on differentiable functions, relative maximums and minimums, applications to the study of functions. Indefinite integrals, integration by parts and by substitution, definite integrals, fundamental theorem of integral calculus, improper integrals. Numerical series, simple and absolute convergence, convergence criteria.
Series of functions, point and total convergence. Complex numbers.
Core Documentation
Marcellini, Sbordone - Elements of Mathematical Analysis 1,2.Marcellini, Sbordone - Exercises in mathematics Vol. 1, 2.
Reference Bibliography
Marcellini, Sbordone - Elements of Mathematical Analysis 1,2. Marcellini, Sbordone - Exercises in mathematics Vol. 1, 2.Type of delivery of the course
Lectures on theory and exercises will be held.Attendance
Attendance is not compulsory but is nevertheless strongly recommended.Type of evaluation
A written test will be held to evaluate the student's ability to use the concepts of integro-differential calculus. An oral test will also be held to evaluate the theoretical knowledge of the course topics.